modules whose surjective ndomorphisms have a γ-small kernels

In this paper, we introduce a proper generalization of that of hopfian modules, called γ-hopfian modules. a right r-module m is said to be γ-hopfian, if any surjective endomorphism of m has a γ-small kernel. some basic characterizations of γ-hopfian modules are proved. we prove that a ring r is semisimple cosingular if and only if every r-module is γ-hopfian. further, we prove that the γ-hopfian property is preserved under morita equivalences. some other properties of γ-hopfian modules are also obtained with examples.